I'm planning to migrate Oracle 10g repository and OMS (both 10.2.0.5) on Windows 2003 32 bit to Oracle Linux 5 64 bit 12C grid control.
My first question is: can I migrate repository to 11g R2 at the same time or should I stick with 10g during migration and after successful migration migrate to 11g R2?

Has any body done this before and would to like to share the steps?
Any documentation available?

We recommend that you upgrade or migrate your repository before you start the EM upgrade process. Since you are doing cross platform db migration make sure that once it’s done the old EM 10.2.0.5 is working fine with the repository on new platform. The EM upgrade using 2-system upgrade approach will take care of the OMS upgrade and migration from one platform to another.

Enterprise Manager 12c live and recorded webcasts (scroll down to see Install, upgrade, agent deployment webcast)
http://www.oracle.com/technology/products/oem/events/webcasts_index.html

Thanks Akanksha.
So cross platform "UPGRADE" is fine?
Can you please elaborate more on when you say "make sure that once it’s done the old EM 10.2.0.5 is working fine with the repository on new platform"?
Also is it OK if I upgrade the 10g repository to 11gR2 in upgrade process?

Yes cross platform upgrade is fine. Once you have done the db upgrade make sure that data is up and running, you can login into the database.. etc to verify that DB upgrade was successfully.
Yes you can do this. Let me share some customer stories with you:
Look at Leighton Nelson ppt : http://www.slideshare.net/leightonnelson